arXiv subjects

Diego A. Rojas

Publications and source records attributed to Diego A. Rojas.

4 recordsLinked to original sources

arXiv2026

Algorithmic randomness in harmonic analysis

Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute characterized Schnorr randomness using an effective version of Carleson's Theorem. We show here that, for computable $1<p<\infty$, the reals at which the Fourier series of a weakly computable vector in $L^p[-π,π]$ converges are precisely the Martin-Löf random reals. Furthermore, we show that radial limits of the Poisson integral of an $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Schnorr random reals and that radial limits of the Poisson integral of a weakly $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Martin-Löf random reals.

math.LO↗

arXiv2025

Effective weak convergence and tightness of measures in computable Polish spaces

Prokhorov's Theorem in probability theory states that a family $Γ$ of probability measures on a Polish space is tight if and only if every sequence in $Γ$ has a weakly convergent subsequence. Due to the highly non-constructive nature of (relative) sequential compactness, however, the effective content of this theorem has not been studied. To this end, we generalize the effective notions of weak convergence of measures on the real line due to McNicholl and Rojas to computable Polish spaces. Then, we introduce an effective notion of tightness for families of measures on computable Polish spaces. Finally, we prove an effective version of Prokhorov's Theorem for computable sequences of probability measures.

math.LO↗

arXiv2021

Effective weak and vague convergence of measures on the real line

We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.

math.LO↗

arXiv2021

Effective notions of weak convergence of measures on the real line

We establish a framework for the study of the effective theory of weak convergence of measures. We define two effective notions of weak convergence of measures on $\mathbb{R}$: one uniform and one non-uniform. We show that these notions are equivalent. By means of this equivalence, we prove an effective version of the Portmanteau Theorem, which consists of multiple equivalent definitions of weak convergence of measures.

math.LO↗